This invention deals with an improved method of three-dimensional seismic imaging which preserves seismic amplitudes, so that the amplitudes on the final seismic image are proportional to the reflectivity of the earth, regardless of the geologic dip, depth of burial, or seismic recording geometry. The technique is easily specialized to two-dimensional DMO, in the case where the shot-receiver axis lies along the direction of survey.
Common-depth-point stacking ("CDP", also known as common-mid-point or common-reflection-point stacking), in which seismic traces from the same surface midpoint but from different shot profiles and having different offset distances are summed to attenuate unwanted signals, is well known in the art. When the subsurface reflector is horizontal, "flat", the established techniques of spherical divergence correction, normal moveout ("NMO") and zero-offset 3D migration produce an accurate 3D seismic image after CDP stacking.
In most practical situations the reflector of interest is not flat. For a dipping reflector an extra step, dip moveout ("DMO"), must be added in order to prevent CDP stacking from attenuating the image of the reflector. The purpose of DMO is to correct finite-offset seismic data to an equivalent zero-offset data set.
There are various alternative equivalent implementations of DMO. Perhaps the most popular are Hale's method and the summation method of Deregowski and Rocca. Hale's Fourier-based method, proposed in his doctoral thesis "Dip Moveout by Fourier Transform" submitted to Stanford University Geophysics Department, May 1983, is carried out in frequency/wave vector (f,k) domain. Deregowski and Rocca's summation method described in "Geometrical Optics and Wave Theory of Constant Offset Sections in Layered Media," Geophysical Prospecting 29, 374-406 (1981), is carried out in time/space (t,x) domain. It involves summation along a "DMO trajectory."
The original work of Deregowski and Rocca was concerned primarily with two-dimensional DMO, in which the line connecting the shot and receiver is co-linear with the direction of the seismic survey line. Hale generalized DMO to the three-dimensional situation, in which the shot-receiver axis can lie in any direction relative to the survey direction. Berg, in "Application of Dip-Moveout by Fourier Transform: Method Overview and Presentation of Processed Data from 2-D and 3-D Surveys," 54th Annual Meeting of the SEG, Atlanta, Expanded Abstracts, 796-799, (1984), showed how to connect Hale's technique with Deregowski and Rocca's summation method. U.S. Pat. No. 4,742,497 to Beasley et al exploited this connection to describe a three-dimensional version of the technique of Deregowski and Rocca. Hale's work had shown that the DMO operation should always be performed along the shot-receiver axis. Beasley et al simply took the summation algorithm of Deregowski and Rocca and executed it along the shot-receiver axis.
It is necessary to do more than simply map each input amplitude along the DMO trajectory, if a true amplitude DMO process is to be achieved. True-amplitude DMO not only puts every event at the correct zero-offset position, but also guarantees that the each event's amplitude is what would have been recorded at zero offset. Kinematic DMO, such as the techniques referred to above, puts each event at the correct space and time position but fails to produce the correct amplitudes. Deregowski and Rocca introduced the notion of convolving the data with a time-variant filter S as part of the mapping process. The key to turning kinematic DMO into true-amplitude DMO is the correct design and application of the filter S.
There have been several prior attempts to turn DMO into a "true-amplitude" process. Deregowski and Rocca experimented with various ad hoc filters S to be applied as part of their summation method but never came up with a
satisfactory solution. In his PhD thesis, Hale unsuccessfully attempted to derive the set of filters in (f,k) space. However, he abandoned these filters in his later published work, "Dip-Moveout by Fourier Transform" Geophysics, 49, 741-757 (1984), and went with unsatisfactory heuristically-derived filters instead. Berg (1985) showed how to transform Hale's heuristic filters into a summation technique similar to Deregowski and Rocca's method, but the results were no better than Hale's.
Recently, Jorden, Bleistein, and Cohen, "A Wave Equation-Based Dip Moveout," 57th Annual Meeting SEG, New Orleans, Expanded Abstracts 718-721 (1987), attempted to connect DMO with the wave equation. They outlined a method for making this connection based upon the Born approximation, but published no details of what kind of DMO filters would result from their analysis. In his doctoral thesis, "Transformation to Zero Offset" submitted to Colorado School of Mines, April, 1987, Jorden outlines a proposal for a seismic imaging algorithm which is related to DMO but is not the same as DMO. His algorithm is applied in the absence of separate spherical divergence and NMO corrections. The concept of using a summation approach with filters is employed, following the approach of Deregowski and Rocca. In addition, the mathematical expressions for the filters are so extremely complicated as to render the algorithm impractical for use in actual surveys. Also, the algorithm is not fully three-dimensional.